const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y term nIn = \x:set.\y:set.~ x iIn y const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoL : set set const SNoS_ : set set const SNoLev : set set const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 var x:set var y:set var z:set var p:set prop var w:set hyp !u:set.u iIn SNoS_ (SNoLev y) -> z = x + u -> SNo u -> SNoLev u iIn SNoLev y -> u < y -> p (x + u) hyp w iIn SNoL y hyp z = x + w claim w iIn SNoS_ (SNoLev y) -> SNo w -> SNoLev w iIn SNoLev y -> w < y -> p (x + w)